 Normal surface coordinates

For $t$ tetrahedra, count the seven disk types in each tetrahedron. Across each glued face the counts of each of its three normal arc types must agree. These are homogeneous linear equations with integer coefficients. An admissible solution also has at most one nonzero quadrilateral type in each tetrahedron. Matching the arcs in their boundary order reconstructs the normal surface up to an isotopy preserving the tetrahedra.